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**A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems. II: Explicit method.**
*(English)*
Zbl 0598.65054

From authors’ summary: In part I [ibid. 11, 277-281 (1984; Zbl 0565.65041)] the authors gave a Noumerov-type method with minimal phase- lag for the integration of second order initial-value problems: \(y''=f(t,y)\), \(y(t_ 0)=y_ 0\), \(y'(t_ 0)=y_ 0'\). However, the method given there is implicit. We show here the interesting result that if the Noumerov-type methods of the paper mentioned above are made explicit with the help of the classical second order method, then there exists a selection of the free parameter for which the resulting method has a considerably small frequency distortion of size \((1/40320)H^ 6\) and also a (slightly) larger interval of periodicity of size 2.75 than the phase-lag of size \((1/12096)H^ 6\) and interval of periodicity of size 2.71 for the implicit method of the authors. More interestingly, it turns out that Noumerov made explicit of the first author [BIT 24, 117- 118 (1984; Zbl 0568.65042)] also has less frequency distortion than the (implicit) Noumerov method.

Reviewer: Z.Jackiewicz

### MSC:

65L05 | Numerical methods for initial value problems involving ordinary differential equations |

34A34 | Nonlinear ordinary differential equations and systems |

### Keywords:

Noumerov-type explicit method; minimal phase-lag; second order initial- value problems; interval of periodicity
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\textit{M. M. Chawla} and \textit{P. S. Rao}, J. Comput. Appl. Math. 15, 329--337 (1986; Zbl 0598.65054)

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### References:

[1] | Brusa, L.; Nigro, L., A one-step method for direct integration of structural dynamic equations, Internat. J. Numer. Methods Engrg., 15, 685-699 (1980) · Zbl 0426.65034 |

[2] | Chawla, M. M.; Rao, P. S., A Noumerov-type method with minimal phase-lag for the integration of second order periodic initial-value problems, J. Comput. Appl. Maths., 11, 277-281 (1984) · Zbl 0565.65041 |

[3] | Chawla, M. M., Numerov made explicit has better stability, BIT, 24, 117-118 (1984) · Zbl 0568.65042 |

[4] | Chawla, M. M.; Sharma, S. R., Families of fifth order Nyström methods for \(y\)″ = \(f(x, y)\) and intervals of periodicity, Computing, 26, 247-256 (1981) · Zbl 0437.34035 |

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